[Computational Methods in Applied Sciences] Advances on Modeling in Tissue Engineering Volume 20 || Cell mechanics: The role of simulation

P.R. Fernandes and P.J. Bártolo (eds.), Advances on Modeling in Tissue Engineering, 1 Computational Methods in Applied Sciences 20, DOI 10.1007/978-94-007-1254-6_1, © Springer Science+Business Media B.V. 2011

Cell mechanics: The role of simulation

Christopher R. Jacobs and Daniel J. Kelly 1

Abstract Computer simulation is one of the most powerful tools available to the applied mechanician to understand the complexities of mechanical behavior. It has revolutionized design of virtually all man-made structures from aircraft and buildings to cell phones and computers. It has also become a relatively important tool in biomechanics and simulation of tissues and implants has become routine. Indeed we appear to be on the verge of patient specific simulation becoming a critical tool in orthopaedic and cardiovascular surgery. However, its use as a tool of basic science is much less clear. In this chapter we explore the potential for mechanical simulation to contribute to improve fundamental understanding of biology. We consider the challenges of creating a model of a mechanobiological system with experimental validation. We propose that the area of cell mechanics is a particular area where simulation can make critically important contributions to understanding basic physiology and pathology and outline potential areas of future advancement.

1 Introduction

was inspired by a Keynote lecture given at the 2007 European Society for Biomechanics Summer Workshop on Finite Element Analysis [1]. In this talk the author reviewed the role of simulation in making insights into biology and focused specifically on the capabilities and limitations of computer modeling to accomplish real hypothesis-testing-based investigation in biology. The contrast was made between hypothesis testing and descriptive research such as showing associations between mechanical and biological parameters or behavior. The conclusion is that one area where simulation might play an increasingly important role in identifying Christopher R. Jacobs Department of Biomedical Engineering, Columbia University, New York, NY, USA, email: [email protected]

Daniel J. Kelly Trinity Centre for Bioengineering, School of Engineering, Trinity College Dublin, Ireland., email: [email protected]

This manuscript is divided into three main sections. The topic of the first section

mailto:[email protected]

mailto:[email protected]

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biologic mechanistic behavior is cell mechanics. Indeed, it is for the most part difficult or impossible to experimentally quantify the mechanical state of a cell, suggesting that simulation can provide a valuable tool, particularly in the area of cell mechanobiology. Mechanobiology being the study of how biological systems sense and respond to mechanical signals in contrast to biomechanics, or the study of the mechanical behavior of biological systems [2]. We begin, in section 2, with a discussion of the potential and limitations of computer simulation of mechanical behavior. With this foundation, section 3 is a mini literature review of current work in cell mechanics, and section 4 is a brief discussion of exciting possibilities for the future.

2 What is the Role of Mechanical Simulation in cell Biology?

numerical simulation of highly complex structures. It allows skilled engineers to accurately predict mechanical response incorporating aspects such as complex geometries, inhomogeneous materials, and complex material behavior such as anisotropy and non-linear behavior. It can also be used to simulate adaptive material behavior such as bone remodeling to specific loading conditions [3, 4]. In addition, modern simulation software has removed many of the time-consuming simulation tasks such as discretization and made simulation available to many more investigators than ever before. However, this ease of use also greatly increases the potential for misuse of the method and the prediction of inaccurate results. Furthermore, these erroneous predictions can appear to be quite authoritative due to sophisticated 3D graphical rendering software.

2.1 Verification, Validation, and Sensitivity

assured through considering three distinct aspects, verification, validation, and sensitivity analysis. The first, verification, involves ensuring that the software being employed is indeed computing a correct solution to the given problem, or correctly solving the equations. In other words, verification is ensuring that there are no errors in programming or bugs in the software package. Often in biomechanics, it is taken on faith that commercial software does in fact converge to a correct solution. Generally this is a safe assumption, however there are some infamous counter-examples [5]. More significant challenges result from issues of capturing singular behavior at non-smooth boundaries, ensuring adequate mesh refinement, non-linear material behavior, and properly dealing with contact behavior. In these cases it is simple to use a commercial code to obtain essentially meaningless results that certainly appear convincing [5] .

Christopher R. Jacobs and Daniel J. Kelly

Anderson, Ellis, and Weiss [5] describe the process by which accuracy can be

The finite element method has emerged as an incredibly powerful tool for


Assuming that the code and modeling strategy are all satisfactory, the next issue is the question of validation. This is the separate question of “to what extent the assumptions and parameters of the model reflect reality.” Many times, in biomechanical analysis, it is not possible to consult a table of material properties as an analyst of traditional engineering structures might. Thus, the solution can only be expected to be a good as the assumptions made and GIGO (garbage in = garbage out) is a serious concern. The best case scenario is when the simulation can be compared to experimental validation data obtained by the same investigator or team [5]. A more common approach is to utilize data published from experi-ments done in another laboratory. The danger with this is that the experimental conditions are often not precisely similar between simulation and experiment, and in many cases the same material behavior parameter can differ wildly depending on experimental conditions. For example, cellular Young’s modulus determined from atomic force microscopy is well known to be extremely sensitive to indenter tip geometry [6]. This implies that to have confidence in simulation results they should be validated with experimental results collected for the model output of interest. Validating a prediction of, for example, displacements do not ensure that force predictions are accurate. Indeed, the traditional finite element method is a displacement-based approach and displacements are accurate to a higher order than derived quantities such as stress or strain.

In reality, we are often faced with situations where one or more model parameters are not known with the precision we would like. This is often the situation with characterization of biological material. In this situation we must conduct a sensitivity or parametric analysis. The simulation can be conducted for a range of values for any unknown parameters, and the sensitivity of output determined. Confidence in the simulation may be gained if the model is insensitive to parameters that can only be determined to fall in some given range. If the simulation results do not change significantly as long as the input parameter falls within this range, there is no problem. In contrast, if the solution depends strongly on a parameter for which experimental data are not available, the specific predictions of the model may not be of much value. However, this situation can also be informative as it can focus experimentation on those parameters that are most critical. The bone remodeling simulation field is an interesting illustration of this principle. Often these simulations incorporate descriptions of biologic, mechanobiologic, and material behavior and interactions that can involved a large number of unknown parameters [7]. However, a parametric analysis suggests that reasonable bone morphologies are predicted regardless of the quantitative formulation adapted [8]. Although this is encouraging in terms of predicting bone behavior, it is also suggestive that it would be difficult to determine the quantitative relationship between bone physiology and mechanics with simulation using current modeling approaches.

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2.2 The Paradox of Validation

validation. As described in 2.1, validation is a critical step creating a high-fidelity mechanical simulation. And validation should be conducted directly on the parameter of interest. In cell mechanics, for example, a simulation might be used to predict the membrane deformation or strain in a cell. This might be of interest to determine whether membrane deformation that occurs with particular physical signal such as fluid shear or substrate stretch is sufficient to activate a particular molecular signaling mechanism such as a stretch activated channel. This model would need to be validated with a measurement of membrane deformation through, for example, optical tracking of membrane-embedded protein. Alternatively, a simulation might be used to estimate the forces across a focal adhesion or even a single integrin [9] when a cell is loaded in a particular way. These predictions might be validated with a direct measurement using, for example, an atomic force microscope. The paradox is that the experimental data collected to validate the model may actually obviate the need for the model. In other words, by the time an experimental system is designed and built to measure cell-level mechanical behavior, it is likely just as easy to use experimental measurement directly than to rely on a model. Particularly since much of the time and money is making cell mechanics measurements is in designing and validating the rig.

A related issue in the application of simulation to cell mechanics is the perceived cost. Often the authors have encountered the assertion that simulation can generate more results than experimentation for a lower cost. However, it may be that this perception is a result of funding models in Europe. Although these models are changing, historically when investigators in certain institutions consider costs they do not necessarily include the cost of students, research staff, or their own time, which may not to be allocated to specific projects. Thus, the cost referred to is predominantly supply expenses, which are of course much lower for simulation. However, in the US model salaries are included in project costs and, indeed, typical represent 80% or more of total cost. This greatly increases the cost of simulation.

Another potential application of simulation in cell mechanics is to critically evaluate mechanical factors that drive biological process at a cell and molecular level and compare the system behavior with experimental data. This would be a great value when experimental strategies alone cannot address these relationships directly as well as to guide and leverage experimental investigations. For example, by comparing the patterns of differentiation in the regenerating tissue within loaded bone chambers with FE predictions of the biophysical environment, Lacroix and Prendergast hypothesized that shear strain and fluid flow regulated differentiation during bone regeneration [10]. It has been possible to continuously test this hypothesis by comparing the predictions of simulations based on the hypothesis with the outcomes of other regenerative events such as fracture healing, osteochondral defect repair and distraction osteogenesis [11-14]. In all cases the same underlying modeling parameters [15] were chosen to simulate tissue differentiation [10, 11, 13-15], so as to avoid the accusation of ‘tweaking’ model


In his presentation [1], the author introduced the concept of the paradox of


parameters to obtain better predictions. These modeling frameworks can also be used to critically compare different hypotheses for mechano-regulated progenitor cell differentiation, by comparing the predictive abilities of the different hypo-theses [16]. Of course, some a priori insight is required to determine the initial set of model parameters. This was described by van der Meulen and Huiskes [2] as one of ‘trial and error’: “Computational mechanobiologists hypothesize a potential rule and determine if the outcome of this hypothesis produces realistic tissue structures and morphologies, hence ‘trial-and-error’. If the results correspond well, they might be an explanation for the mechanism being modeled.”

There are limitations to the above approaches. For example, models of mechanical regulation of tissue differentiation [10, 17] suggest that certain levels of shear strain and fluid flow acting on cells in a regenerating tissue will promote fibro-cartilaginous tissue formation. To date, these models have not been able to elucidate whether these cellular stimuli promote differentiation directly, or whether, for example, the mechanical environment is acting indirectly to promote chondrogenesis by inhibiting angiogenesis and thereby promoting the formation of a hypoxic environment known to promote chondrogenesis. Explicitly modeling the process of angiogenesis within these modeling frameworks may allow such questions to be more critically addressed [18].

Similar modeling approaches have been used in diverse areas of orthopedic science, from understanding the role of mechanics in development [19] to tissue organization during regeneration [20]. However, when comparing candidate cell-level mechanical signals it is important to remember the bone remodeling experience ([8] and 2.1 above) and critically evaluate the discriminatory power of comparing simulation results with histological or morphological data. Using a bone-ingrowth chamber Prendergast and colleagues have found that in order to evaluate falsifiable hypotheses concerning cell-level mechanical signals that regulate tissue differentiation a much larger set of animal-specific model with a high level of variability need to be considered than have been utilized in the past [21]. This is suggestive that perhaps important aspects of mechanoregulation in biology can be discerned with this approach when careful consideration is given to forming testable hypotheses, awareness of the relevant cell and molecular biology, and a critical level of biofidelity is achieved in the modeling.

3 State-of-the-art in Computational Cell Mechanics

mechanical behavior of cells, there have been important advances as well. In this section we review the recent literature regarding computational models of cell mechanics. The fundamental challenge that faces all of these efforts is how to approach the exceedingly complex internal structure of the cell and the diversity of behaviors which depend on cell type and morphology. We have grouped the review into homogenization approaches, microstructural models of the cytoskeleton, microstructural models of the cell membrane, and microstructural models of the

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Although there are important limitations and pitfalls to consider in simulating the

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entire cell with a focus on multi-scale modeling. By its nature this mini-review will only highlight important aspects of the field. More details treatments are already available in the literature [22].

3.1 Homogenization

microstructure of the cell is treated in a spatially averaged way with the goal of describing its apparent level behavior. Thus, these approaches include lumped-parameter and rheological models here. These approaches have been particularly valuable in the analysis and interpretation of local measurements of cellular behavior such as atomic force microscopy (AFM) and magnetic twisting cytometry.

Although simple in nature, treating red blood cells as elastic membranes surrounding a Maxwell viscous fluid can make surprisingly accurate predictions for the fluid properties of blood [23]. They have also been applied to under-standing of leukocytes mechanics and adhesion [24] where they have been expanded to allow for large deformations [25]. This problem is made particularly challeng-ing by the need to account for complex boundary conditions. For example, sliding must be included to simulation micropipette experiments needed for validation, but non-linear adaptive adhesion is required to simulate interaction with the vessel wall. The latter process is particularly complex and simulations have included effects of receptor/ligand density and affinity as well as cell motion and motility on the adhesive contact mechanics in both discrete and continuum approaches [26, 27]. Interestingly, it is known that the receptors that mediate extracellular adhesion, e.g. integrins, tend to function in clusters rather than discrete units [28]. The resulting simulation is highly nonlinear and involves two levels of coupling, that between cell deformation and adhesion as well as the fluid-structure inter-action between cell deformation and fluid shear stress [25, 29-31].

Cell mechanics simulations have also been important in understanding structural tissues (i.e. tissues whose primary function is supporting or generating load). Simulations of smooth muscle cells have allowed investigators to make predictions of internal stress and strain [32]. The mechanical behavior of chondro-cytes has been simulated with a multiphasic constitutive model. In this approach the viscoelastic material properties of the cell are deduced from the mechanical interaction of a hyperelastic cytoskeleton and the fluid cytoplasm flowing past it [33]. Chondrocyte models have been directly validated with AFM measurements and, interestingly, investigators were able to infer the mechanical behavior of the peri-cellular matrix as well as the cell itself [34].

An enclosed viscoelastic Maxwell fluid has also been utilized to simulate the mechanics of the nucleus [35]. However, although the cell’s plasma membrane can be considered to be a simple elastic membrane with negligible bending stiff-ness, the nuclear membrane exhibits more complexity. The results of micropipette experiments on isolated nuclei have been accurately simulated with a three-layered nuclear membrane consisting of one elastic layer and two viscoelastic layers modeled rheologically as a standard linear solid.


For our purposes we define homogenization approaches to be those in which the


Despite their success in blood cells and chondrocytes, homogenizing the cytoplasm/cytoskeleton fails to reflect the complex microstructure of the cyto-skeletal polymer network that is important for many cell types. Critical structures such as the actin cortex or stress fibers are intracellular mechanical structures that are crucial to the cells mechanical performance. Of course, it would be impossible to fully review polymer network theory here, however, the diversity of cytoskeletal mechanics are of particular interest. For example, the polymers of the cytoskeleton imply that a single theory or approach is unlikely to yield a sufficient description. For example, the persistence length of microtubules is greater than a typical cell diameter. Thus, solid mechanics continuum models (e.g. simple columns or beams) may be appropriate. On the other hand, entropic effects can be responsible for the forces in actin microfilaments and intermediate filaments. However, when actin is highly cross-linked, its entropic behavior may be limited and a continuum description is again appropriate.

A homogenization strategy for cytoskeletal networks involves selecting a set of apparent level parameters to describe the network microstructure. For example volume fraction, orientation, and level of cross-linking might be appropriate. These parameters are then linked to apparent-level constitutive material behavior. Kwon et al. [36] presented a computational paradigm for achieving this homogenization and demonstrated it for actin networks of varying densities and anisotropies. The cytoskeleton has also been modeled as a fiber-reinforced composite and the predictions validated with whole-cell data from AFM and magnetic twisting cytometry [37]. Kamm and colleagues have recently presented a Brownian dynamics simulation of actin and actin cross-linkers to demonstrate regimes of entropy-dominated and energy-dominated behavior depending on polymer and crosslinker density as well as pre-stress [38].

3.2 The Cytoskeleton

mechanics and cytoskeleton-mediated processes such as motility and force generation, but also because the cytoskeleton is a site of mechanosensation [39] and the level of cytoskeletal pre-tension may regulate the cell’s response to mechanical loading [40-42]. Microstructural models of the cytoskeleton often treat polymers as linear structures that can resist axial loads only. In the literature these systems are often referred to as tensegrity structures [43], although strictly speaking this is incorrect. Tensegrity systems incorporate only compression-only “struts” and tension-only “cables” and, in cells, microfilaments often supports compression and microtubules are frequently loaded in tension. Nonetheless, tensegrity models of the cytoskeleton have been shown to predict some of their dynamic properties [44]. A complete review of the diverse strategies for modeling cytoskeletal mechanics is presented in [45]. Additionally, microstructural models of the cytoskeleton are commonly a component of whole-cell microstructural models which will be discussed in the section 3.4. Intergrin mediated attachment

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The mechanics of the cytoskeleton are not only important in understanding cell

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of the cytoskeleton to the extracellular environment can also be critical to under-standing cytoskeletal mechanics, not only as it relates to mechanical coupling, but also in terms of sensing mechanical loads and in the context of pathologies such as cancer [46].

3.3 The Cell Membrane

than the cytoskeleton. Perhaps this is a result of a perception that the role of the cytoskeleton is primarily structural while the role of the membrane is primarily as a barrier. Indeed, this may be the case for the lipid bilayer. Due to the high mobility of the phospholipids that make up the bilayer, the shear modulus within the plane of the bilayer is negligible. As a consequence, its bending stiffness is also nearly zero. This accounts for the success of using the two dimensional membrane equation to describe its mechanics such as models described in section 3.1.

The low resistance of the bilayer to bending does, however, produce some interesting in-plane behavior. In many cell types and conditions, the cell’s bilayer is slack state with in-plane stresses that are nearly zero. In this situation the membrane tends to take on an undulating configuration with a large number of folds or wrinkles. Under entropic Brownian influences, these folds can actually allow an effective or apparent tension to develop in the bilayer. Furthermore, this effect allows the bilayer to resist in-plane strain. Thus, the elasticity described in many membrane models originates entropically rather than from inter-molecular forces as in classical continuum mechanics. A separate mechanism of bilayer force generation occurs when it is stretched to the point that all of the folds have been removed. In this situation the hydrophobic interactions between the phospholipids resist further stretching and produce highly nonlinear behavior that approaches inextensibility. Indeed, the bilayer failure strain is on the order of 3%.

Although the bilayer had negligible bending stiffness, this is not necessarily true of the cell membrane. Often an underlying cytoskeletal network supports the bilayer. In red blood cells this is a specialize protein known as spectrin. In other cells actin provides bending resistance to the membrane. The resulting composite structure can have complex mechanical behavior including a significant trans-verse bending stiffness. A number of different continuum frameworks have been employed to describe cell membrane mechanics in simulations including classical shell theory, fluid mechanics, and statistical mechanics (reviewed in [47]). The mechanical interaction of the cytoskeleton and bilayer has been simulated in the context of actin polymers advancing the membrane during the extension of neurons. Also, since the since the bilayer is functionally a two dimensional fluid and only anchored to the member at discrete points, it is possible to pull cylindrical bilayer extensions known as tethers from a cell with functionalized micropipettes or AFM tips [48, 49]. Simulations of such processes can provide important validation of cell mechanics models.


The cell membrane has perhaps received less attention as a mechanical structure


3.4 Whole Cell Models and Multiscale Approaches

cytoskeleton have become more sophisticated, they have begun to be more frequently integrated to predicted whole cell mechanical behavior. For example, a whole-cell model of fibroblasts in suspension has been shown to predict optical tweezers data when details of the cortical actin are included [50]. Quantitative cell-cell forces have been determined in embryonic epithelial cells using a system of orthogonal dashpots to represent the viscoelastic cytoplasm/cytoskeleton [51]. Membrane blebbing (for formation of round protruding folds of membrane) has been simulated in whole-cell models that treat the membrane and cytoskeleton as elastic structures and an operator splitting approach to treat the membrane-cytoplasm fluid-structure interaction problem [52]. The bundles of cilia that act as mechanosensors in hair cells have been simulated in detailed finite element models that include cytoskeletal proteins, anchoring mechanics, bundle cross-linking, and tip links [53, 54]. Even the anisotropic deformation of yeast under extreme hydrostatic pressures (over 200MPa) have been simulated [55]. One challenge in whole-cell models is the complex microstucture of the cytoskeleton. The effect of the cytoskeleton has been incorporated into whole-cell models by simplifying its complex microstucture into a set of representative skeletal elements [56, 57]. This approach has also been applied to simpler spectrin network of the red blood cell [58].

Another strategy for microstructural simulation of entire structures is direct treatment of the microstructure. Highly efficient computational approaches have allowed for very large scale simulations of on the order of hundreds of millions of structural components. Such simulations have proven highly effective for treating microstucturally complex materials such as trabecular bone [59-61]. However, such approaches are insufficient to tackle the complexity of the cell at the filament or even fiber bundle level. As a result the only option for conducting whole-cell simulations with highly fidelic microstructures is to take a multi-scale approach. Although still in its infancy, some cell multi-scale simulations have been conducted. For example, Hartmann et al. recently simulated the deformation of a red blood cell deformed with optical tweezers with a multi-scale approach [62].

4 Future Directions

more and more on multi-scale simulations. Interestingly, for cell mechanics this need not be limited to a simple matter of course-graining due to the size of the problem. In cell mechanics the challenge is inherently one of multi-physics as well. There is the obvious coupled fluid-structure problem represented by the cytoplasm and cytoskeleton/membrane respectively. However, other multi-physics issues are associated with adhesions and the cellular response and adaptation to mechanics as well as other phenomena not yet considered. This raises the potential

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As microstructural models of cellular components such as the membrane and

Of course, we can expect future simulations of cell mechanical behavior to rely

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that the ideal computational architecture for the different aspects of the problem might be quite different, as they are, for example, in fluid versus solid mechanics. Thus the application of heterogeneous computing approaches may play a critical role in the future of cell mechanics simulation. Indeed, cell mechanics could become one of the grand challenge problems in this new and exciting field.

Although multi-scale, multi-physics, and heterogeneous computing may be critical to future whole-cell mechanics simulations, there is also great potential for insight into fundamental cell behavior by modeling specific parts or components of cells. Models of the nucleus (described above) have led to insight into its architecture and mechanical behavior. The hair-cell cilia example described above is another example. Multi-cilia in the respiratory and female reproductive tracts are another. In this case the cilia are motile and molecular motors cause them to exhibit a wave-like beating motion that mobilize the extracellular fluid. Recently, computer simulations have been utilized to describe a fluid-structure coupling between multi-cilia and the bathing fluid that appears to be central to maintaining the coordination and progression of travelling wave [63].

Another area where very simple simulation may lead to important insights is in the mechanics of primary cilia. Primary cilia are related to multi-cilia and haircell cilia. However, they are nearly ubiquitous (only certain blood and kidney cells are known not to form primary cilia), every cell only has a single one, and until the last decade their function was largely a matter of speculation (Figure 1). However, recently, a rapidly increasing body of evidence suggests that primary cilia act as extracellular sensors, sampling the biomechanical and biochemical environment some distance away from the cell [64, 65]. In our laboratory we are focused on understanding the ability of primary cilia to sense fluid flow. Specifically, this is likely related to the deformation that the cilium experiences. We have recently constructed a simple-but-effective large-rotation beam-bending model of the cilium that allows for rotation of the intracellular anchorage within the cilium (Figure 2). Interestingly, when rotation was not allowed the predicted flexural rigidity of the model was half the value obtain when rotation was allowed.

Multi-scale modeling may also overcome many of the limitations that have impeded the ability of simulation to truly accomplish hypothesis-testing investigation. We previously referred to the bone remodeling simulation field, and the observ-ation that reasonable bone morphologies could be predicted for many quantitative formulations (7). By combining different hierarchal levels of experimental data, it may be possible to use multi-scale modeling to ask not if different magnitudes of stress, strain or strain energy density at the tissue level regulate bone remodeling, but rather if alterations in tissue level loading can lead to deformation in specific cellular structures or even to conformational changes in particular proteins. In this way it may be possible for modelers to explain how the observations of experimentalists at the cell or molecular level can result in changes observed at a tissue level during development, aging and disease.



Fig. 1: Primary cilia in cultured kidney cells imaged by confocal microscopy (left) and epifluorescence (right).

Fig. 2: We validated our simple model of primary cilium bending with taken from the literature for kidney cells in culture [66]. Interestingly, when basal twisting is not allowed (left) the predicted flexural rigidity is half as large as when it is allowed (right).

5 Conclusions

biomechanics today. It is at the heart of a number of diseases responsible for great human suffering. It is also central to understanding cellular mechantransduction (particularly in non-excitable cells) and the mechanobiology of structural tissues such as bone and cartilage. Yet despite this critical importance, it receives dramatically less attention than traditional biologic and biochemical research. Perhaps this is due to the scarcity of individuals trained with deep understanding

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The mechanical behavior of cells is one of the most significant challenges facing

in both engineering mechanics simulation and modern cell and molecular biology.

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The authors would like to strongly encourage work in this area and particularly new investigators consider the potential for critical new insight by pursuing this interdisciplinary field. Cell mechanics represents not only one of the highest-impact areas the young biomechanical engineer can choose to work in, but also one of the most demanding challenges requiring facility in the state-of-the-art in mechanical simulation technology.

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[Computational Methods in Applied Sciences] Advances on Modeling in Tissue Engineering Volume 20 || Cell mechanics: The role of simulation